A Reconstruction of the Differential Analyzer in Meccano

FEATURE A Reconstruction of the Differential Analyzer in Meccano The challenges of torque amplifiers, integrators, and backlash By Tim Robinson was first introduced to Meccano, a child’s educational construction set cre- ated in the United Kingdom, at about the age of six and quickly became fascinated with it as a medium for constructing working mechanisms and small machines. Over the next ten years or so, I gath- Iered quite a large collection. I first attempted to construct a differential analyzer in Meccano around 1971. I had just encountered calculus in high school and at the same time I started to develop an interest in computers. One of the first books I read on computers included a chapter on analog computation, and on the differential analyzer in particular. Significantly, the book briefly men- ©DIGITALVISION tioned that simple differential analyzers had been constructed in Meccano in the 1930s. So began an interest that has remained with me for more than 30 years. Early Attempts My early attempts were not very successful because of the difficulty of constructing functioning torque amplifiers. When Hartree and Porter built the first Meccano differential analyzer at Manchester Universi- ty, their goal was to build a working machine quickly. They used Meccano simply because it was readily available and allowed them to avoid designing and custom machining most of the required parts. However, Hartree and Porter felt no constraint to stay within the limits of the Meccano system. In particular, they did not believe that adequate torque amplifiers could be made without custom machining. In contrast, as a Meccano enthusiast approaching the same subject, finding a solution within the system to what at first seems an insurmountable problem was part of the challenge. 0272-1708/05/$20.00©2005IEEE 84 IEEE Control Systems Magazine June 2005 half of which are nuts, bolts, and washers. Construction started about four years ago, after a flash of insight led me to a robust solution to the amplifier problem. Instead of arranging the input and output shafts of the amplifier to be colinear with one drum on each as Bush had done, I arranged the input shaft to run parallel to the output, with both drums running on the output shaft (Figure 2). The input is coupled to the input arms of the amplifier by means of a pair of gears, one fixed on the input shaft and the other free to rotate on the output shaft between the drums. Since the output shaft has the benefit of the amplifi- cation of torque compared to the input, any residual friction from the drums running directly on the output shaft is unimportant. While this feature was the key to constructing a working amplifier within the constraints of the Meccano system, another critical element turned out to be the Figure 1. Aerial view of the machine. Down the center is the choice of material for the friction bands. Particularly in the interconnect section in which shafts and gearing are placed second stage, the bands are under heavy load and can heat to “program” the machine for a particular equation. On the up significantly from the friction. If the band material is not left are the input table and dual output table, and on the right stable, the critical adjustment of tension in the bands will are four integrators. Each of the functional units can be be upset during operation. If the material stretches, exces- detached from the central interconnect for maintenance and transportation. (Photo: Michael Baxter.) sive backlash is introduced and the bands are likely to slip off the drums. If the material shrinks, then the bands tight- en, increasing friction further in a cycle of positive feedback Having failed in that first attempt, I revisited the differen- that leads to the amplifier seizing up. After experimenting tial analyzer periodically. The main obstacle to making the with many materials, I finally selected Dacron cord, a mate- amplifier is the lack of any form of concentric shafting sys- rial favored by kite flyers because of its light weight, great tem. In Bush’s prototype machine, the amplifiers use drums strength, and resistance to stretching. Dacron performs that rotate on hollow shafts rigidly fixed to the framework. extremely well for my torque amplifier design. This arrangement allows the input and output shafts to pass With these fundamental issues addressed, I evaluated a through with no friction coupling directly from the drums to number of design variations, varying in minor details most- the input. Copying the general layout in Meccano, but without ly to improve strength and rigidity. I steadily improved the the fixed sleeve between the input shaft and performance and finally arrived at a two- the drum, inevitably means that an unac- stage design that runs reliably for days at a ceptable torque is coupled from one of the time (Figure 3). The rest of the machine is drums directly to the input shaft. This cou- relatively straightforward by comparison. pling causes the integrator wheel to steadily slip in the direction of the drum’s rotation. Integrators By adding a counter-rotating element to the Integrators are the principal functional input shaft to at least partly compensate for units of a differential analyzer, and a few the undesirable friction, I succeeded in mak- details of the integrators in the model are ing a simple two-integrator demonstration worth highlighting (Figure 4). The output model. Although the amplifiers had only a of an integrator is taken from the integrat- single stage, with the use of ground glass ing wheel, which rests on the disk under instead of plain glass discs to increase the very light pressure. The available torque friction available at the integrator wheel, the is further reduced by any vibration when mechanism could be made to solve the sec- the machine is in operation. To ensure Torque amplifier ond-order equation for simple harmonic Figure 2. there is no slipping, friction on the shaft stage. The output shaft to the motion at least qualitatively. that couples the integrating wheel to the left carries both rotating drums. The input shaft runs input of the torque amplifier must be mini- A Successful Torque parallel on the right, carried in mal. Friction reduction is accomplished Amplifier rotating bearings, and couples by mounting the shaft in rotating bearings, The machine I have today (Figure 1) con- to the input arms through a driven from the output of the torque tains about 16,000 individual parts, about pair of gears. amplifier. Because the bearings rotate at June 2005 IEEE Control Systems Magazine 85 the same rate as the shaft, friction is essentially reduced to Backlash zero. If the integrator wheel is lifted clear of the integrator In all mechanical systems, there is unavoidable backlash. disk and set in motion, the wheel will spin almost indefi- This fact is particularly true in the case of Meccano, which, nitely. Although the rotating bearings introduce the possi- being essentially a toy, is not manufactured to especially bility of feedback-path coupling from the output of the close tolerances. Starting from an integrating wheel, there amplifier back to the input, there appears to be no evi- is backlash in the torque amplifier as well as in the gears in dence of instability in practice. the shafts connecting the output of the amplifier to either a The integrator disks are made from window glass cut to lead screw or the disk of another integrator. In an analog a circle with a diameter of 11 in and attached to a standard machine, where variables are represented by shaft angles, Meccano component using hot glue. Glass is a convenient, backlash represents a significant source of error. In his readily available material that provides a hard-wearing, prototype, Bush introduced something he called a “front- flat surface. A large diameter provides greater accuracy, and lash” unit to compensate the backlash to first order. These this size was chosen because the longest Meccano units took the form of epicyclic gear assemblies, which screwed rod, used for the carriage lead screw, is 11-1/2 in. could be inserted as required in the interconnect. Whenever The disk is fairly heavy and must rotate very freely. To the direction of motion of the input to the frontlash unit facilitate free rotation, the builders of the early Meccano reverses, a small amount of extra motion is added to the differential analyzers again reached outside the system output to compensate for the motion lost to the backlash. and used commercial ball bearing assemblies to carry the The frontlash unit proved to be another difficult item to disk. I adopted an effective alternative solution, namely, model in Meccano, not because of any problem in princi- supporting the weight of the disk on three small wheels ple but because of the difficulty of creating a unit that can positioned near the periphery. The driving shaft provides operate smoothly under the heavy loading of actual opera- centering and the driving torque, but does not need to sup- tion. After a number of unsatisfactory attempts, I solved port the weight. The disk can be lifted off for cleaning. this problem by using a differential-based design, located The integrator carriage must be accurately positioned to between the two stages of each torque amplifier. By posi- represent the value of the function being integrated. Any play tioning the differential there, the frontlash unit operates in the lead screw directly affects the accuracy of the machine. under only a light loading since it is followed by the gain of To avoid play, two nuts are used on the screw, forced apart the second amplifier stage.

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